Mathematics

# $\displaystyle \int {{x^3}{e^{{x^2}}}dx = }$

$\dfrac{1}{2}\left( {{x^2} - 1} \right){e^{{x^2}}} + c$

##### SOLUTION
C)  $\int x ^ { 3 } e ^ { x ^ { 2 } } d x$

put $x ^ { 2 } = t \Rightarrow 2 x d x = d t$

$\dfrac { 1 } { 2 } \int t e ^ { t } d t = \dfrac { 1 } { 2 } \left[ \left( t e ^ { t } \right) - \int e ^ { t } d t \right]$                {integration by parts}

$= \dfrac { 1 } { 2 } \left[ t e ^ { t } - e ^ { t } \right]+c$

$= \dfrac { 1 } { 2 } \left( x ^ { 2 } e ^ { x ^ { 2 } } - e ^ { x ^ { 2 } } \right) + c$

$= \dfrac { 1 } { 2 } \left( x ^ { 2 } - 1 \right) e ^ { x ^ { 2 } } + c$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
If $\displaystyle I = \int \frac {e^{3x}}{1 + e^x} dx$, then I equal
• A. $\displaystyle (1/2)(1 + e^x)^2 - (1/3)(1 + e^x) + \log (1 + e^x) + C$
• B. $\displaystyle (1/2)(1 + e^x) (e^x + 3) + \log (1 + e^x) + C$
• C. $\displaystyle (1/2)(1 + e^x)^2 - 2 \log (1 + e^x) + C$
• D. $\displaystyle (1/2)(1 + e^x)(e^x - 3) + \log (1 + e^x) + C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
$\int { \cfrac { dx }{ \sqrt { 16-4{ x }^{ 2 } } } } =$?
• A. $\cfrac { 1 }{ 4 } \sin ^{ -1 }{ \left( \cfrac { x }{ 2 } \right) } +C$
• B. $\cfrac { 1 }{ 2 } \sin ^{ -1 }{ \left( \cfrac { x }{ 4 } \right) } +C$
• C. none of these
• D. $\cfrac { 1 }{ 2 } \sin ^{ -1 }{ \left( \cfrac { x }{ 2 } \right) } +C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Hard
Let $\displaystyle \:f\left ( x \right )$ be a continuous function such that $\displaystyle \:f\left ( a-x \right )+f\left ( x \right )= 0$ for all $\displaystyle \:x \epsilon \left [ 0, a \right ].$ Then $\displaystyle \:\int_{0}^{a}\frac{dx}{1+e^{f\left ( x \right )}}$ is equal to
• A. $a$
• B. $\displaystyle \:f\left ( a \right )$
• C. $\displaystyle \:\frac{1}{2}f\left ( a \right )$
• D. $\displaystyle \:\frac{a}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle \lim_{n \rightarrow \infty} \left[\displaystyle \frac{\sqrt{n^{2}-1^{2}}}{n^{2}}+\frac{\sqrt{n^{2}-2^{2}}}{n^{2}}+\frac{\sqrt{n^{2}-3^{2}}}{n^{2}}+\ldots.n terms\right]=$
• A. $\displaystyle \frac{\pi}{2}$
• B. $\displaystyle \frac{\pi}{3}$
• C. $\displaystyle \frac{2\pi}{4}$
• D. $\displaystyle \frac{\pi}{4}$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.